A near-wall variable-Prandtl-number turbulence model is developed for the calculations of high-speed compressible turbulent boundary layers. The model is based on the k-e and the 0 2 -e bk = coefficients in the expansion for k + in the near-wall region <*uv»b uv = coefficients in the expansion for ~uv + in the near-wall region _ #v0> bve = coefficients in the expansion for vO + in the near-wall region a 2 , b 2 = coefficients in the expansion for 6 + 2 in the near-wall region #e0> b eQ = coefficients in the expansion for e# + in the near-wall region B-constant in law of the wall Qx = model constant taken to be 0.1 C d i = model constant taken to be 1.8 for boundary layers and 2.0 for internal flows Cd2 = model constant taken to be 0 Qo = model constant taken to be 0.72 Q 4 = model constant taken to be 2.2 C ds = model constant taken to be 0.8 Cf = skin-friction coefficient, 2r w /(pU 2 X) ) Ch = heat transfer coefficient, /[(poot/ooC^ -e r )] C d = model constant taken to be 1.5 C e2 = model constant taken to be 1.83 C M = model constant taken to be 0.096 C x = model constant taken to be 0.11 f Wt 2 = near-wall damping function for e equation f Wtf e = near-wall damping function for e e equation fp = near-wall damping function for turbulent momentum diffusivity /x = near-wall damping function for turbulent heat diffusivity H = instantaneous total enthalpy, C P T + l /2U k U k h = half-channel width or pipe radius k = turbulent kinetic energy k + = normalized k, k/u 2 M = Mach number M t = local Mach number, u r /(yRT w ) y2 P$ = production due to mean temperature, defined as -(ue)(d(Q)/dx) Pr = molecular Prandtl number *Pr t = turbulent Prandtl number p -instantaneous pressure q w = heat flux at the wall R = univer...

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At present, it is not clear how the fluctuating temperature at the wall can be properly specified for near-wall turbulent heat-flux models. The conventional approach is to assume zero fluctuating temperature or zero gradient for the temperature variance at the wall. These are idealized specifications and the latter condition could lead to an ill-posed problem for fully developed pipe and channel flows. In this paper, the validity and extent of the zero fluctuating wall temperature condition for heat transfer calculations are examined. The approach taken is to assume Taylor series expansions in the wall normal coordinate for the fluctuating quantities that are general enough to account for both zero and nonzero temperature fluctuations at the wall and to develop a near-wall turbulence model allowing finite values of the wall temperature variance. As for the wall temperature variance boundary condition, it is estimated by solving the coupled heat transfer problem between the fluid and the solid wall. The eddy thermal conductivity is calculated from the temperature variance and its dissipation rate. Heat transfer calculations assuming both zero and nonzero fluctuating wall temperature reveal that the zero fluctuating wall temperature assumption is quite valid for the mean field and the associated integral heat transfer properties. The effects of nonzero fluctuating wall temperature on the fluctuating field are limited only to a small region near the wall for most fluid/solid combinations considered.

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